The convergence of nonnegative solutions for the family of problems −Δpu=λeuasp→∞
Mihai Mihăilescu, Denisa Stancu‐Dumitru, Csaba Varga · ESAIM Control Optimisation and Calculus of Variations · 2017
LetΩ⊂ ℝN(N≥ 2) be a bounded domain with smooth boundary. We show the existence of a positive real numberλ*such that for eachλ∈ (0,λ*) and each real numberp>Nthe equation −Δpu = λeuinΩsubject to the homogeneous Dirichlet boundary condition possesses a nonnegative solutionup. Next, we analyze the asymptotic behavior ofupas p→∞and we show that it converges uniformly to the distance function to the boundary of the domain.